Pure Mathematics

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For introduction and overview see Philosophy of Mathematics. For applications see Applications of Mathematics.

Contents

Branches of Pure Mathematics

See also foundations. Algebra, Number Theory and Combinatorics are related to discrete mathematics, i.e. mathematics dealing with objects that can assume only discrete separated values, and thus expressed with integers. Contrast with e.g. Real Analysis (Calculus) dealing with real numbers.

Algebra

The abstract study of number systems and operations within them.

Arithmetic

The concepts of natural number and integer, binary operations of addition and multiplication; theory of divisors, fractions, rational numbers, and irrational numbers. Number systems and notation. Calculation with decimals, divisibility rules, calculation of square, cube, and higher roots.

(Number Theory could be considered a branch of Arithmetic, sometimes called Higher Arithmetic.)

Elementary and multivariate algebra

Algebra as an extension of arithmetic, addition, subtraction, multiplication, division, root extraction, complex numbers, geometric representation of numbers. Polynomials and rational function, solutions of equations, extraction of roots of a polynomial, common roots of two polynomials.

Linear and multilinear algebra

Vector spaces, matrices, linear transformations and linear operators, linear functional, inner product and inner product spaces.

Abstract Algebra / Algebraic structures

Branches include commutative algebra, representation theory, homological algebra, universal algebra. There are a number of algebraic structures: Lattices, groups, fields, rings, categories,

(Note that Linear algebra, elementary number theory, and discrete mathematics are sometimes considered branches of abstract algebra. I.e. Abstract Algebra assumes a greater remit. Ash (1998) includes the following areas in his definition of abstract algebra: logic and foundations, counting, elementary number theory, informal set theory, linear algebra, and the theory of linear operators.)

Number theory (Higher Arithmetic)

Theory of the positive integers. It is based on ideas such as divisibility and congruence. Its fundamental theorem states that each positive integer has a unique prime factorisation. In some ways it is the most accessible discipline in pure mathematics for the general public: for instance the Goldbach conjecture is easily stated (but is yet to be proved or disproved). Is concerned with Elementary number theory, simple arithmetic operations with integers, prime numbers, the sieve of Eratosthenes, Fermat and Mersenne primes, Bernouilli numbers and Fermat’s last theorem, quadratic residues, continued fractions.

Notation

Algebraic number theory, Gauss, Kummer, and Dirichlet, concepts fo field and ring, unique factorisation and ideals; class group and class number; units and their properties; Abelian fields

Analytical number theory, additive and multiplicative properties of numbers, distribution of prime numbers, application of the Circle method, unsolved problems.

Geometric number theory, Lagrange and Seeber, convex body; defnitions and properties of a star body; theory of packings and lattice packign by convex bodies; nonhomogeneous problems, Euclid’s algorithm

Probabalistic number theory.

Combinatorics and combinatorial geometry

Concerned with ‘arrangements, operations, and selections within a finite or discrete system.’

History

Methods, results and unsolved problems, e.g. problems of enumeration, permutations, and combinations, the necklace robelm and Plya’s theorem, the Mobius invesion theorem, special problems of enumeration; problems of choice, König’s theorem, Ramsey’s theorem. Deisng, latin squares, arrays, and coding. Balanced imcomplete block designs, Graph theory.

Combinatorial geometry

Geometry

Euclidean geometry

Euclid’s work and his Elements. Geometry as an abstract discipline, with axioms of order, incidence, congruence, parallels, and continuity. Using deductive method. Measure of polygons and polyhedra. Transformation of geometry. Geometric constructions, ruler and compass constructions. Geometry of more than three dimensions. Concept of convexity and convex sets.

Non-Euclidean geometry

Notion that Euclid’s geometry is not the only possible model of physical space. Particularly the challenge to Euclid's fifth postulate, the parallel postulate by Hyberbolic and elliptic geometry. Development of hyperbolic geometry by Bolyai and Lobachevsky, and elliptic geometry by Schläfli, Riemann, and Klein.

Albert Einstein's general theory of relativity describes space as generally flat (i.e., Euclidean), but as elliptically curved (i.e., non-Euclidean) near regions where energy is present. This kind of geometry, where the curvature changes from point to point, is called Riemannian geometry. Applications of Mathematics#Riemannian geometry

Projective geometry

Seminal work of Pappus of Alexandria and Desargues, and development by Poncelet and others. Perspective projection, points of infinity, concept of ideal points. Projective theorems: Desargues’ theorem and Pappus’s theorem. The concept of duality. Homogeneous coordinates.

Analytic and trigonometric geometry

Plane analytical geometry. Cartesian coordinates. Representation of straight linse by means of algebraic equations. Conic sections, circle, ellipse, yberbola, and parabola. Tangents and normals to curves. Trigonometry as a computational science to calculate angles and distances in plane and spherical triangles. Trigonometric functions, degrees and radians, the sine, cosine, tangent, cotangent, secant, and cosecant functions. Plane trigonometry, spherical trigonometry, analytical trigonometry. Polar coordinates, barycentric and areal coordinates. Special curves: the folium of Descartes, the lemniscate of Bernoulli, the cardioid, the cycloid, the catenary, the brachistochrone.

Combinatorial geometry

See combinatorics.

Differential geometry

Quadratic differential forms, manifolds, tensor bundles, operations on tensor fields, connections. De Rham and Hodge theorems. The Gauss-Bonnet formula. Elliptic differential operators. Isometric imbedding, submanifolds. Complex manifolds

Algebraic geometry

Algebraic varieties in projective space

Analysis

Deals with continuous change and with certain general types of processes that have emerged from the study of continuous change, such as limits, differentiation, and integration. Since the discovery of the differential and integral calculus by Isaac Newton and Gottfried Wilhelm Leibniz at the end of the 17th century, analysis has grown into an enormous and central field of mathematical research, with applications throughout the sciences and in areas such as finance, economics, and sociology.

Real Analysis (Calculus)

Real analysis which includes the analysis of number systems and their properties (rational and irrational numbers, complex numbers, quaternions, transcendental numbers, infinite cardinal numbers infinite ordinal numbers).

Functions, including Fourier series, and differential calculus including the use of derivatives to determine maxima and minima of functions.

Measure with study of problems of determining the length of a curve, area of surface or volume of solids. Integral calculus: definitions of the Riemann, or definite, integral functions, fundamental theorem of calculus.

Complex Analysis

Complex analysis which includes theory of analytical functions of one or several complex variables. Cauchy theory, meromorphic functions, elliptic functions, holomorphic functions. Conformal maps, complex manifolds, cohomology of an n-dimensional complex analytica manifold, complex line bundles, Riemann-Roch theorem, Oka’s theorems, Runge’s theormes, The Cousin problems, automorphism.

Potential theory including harmonic, polyharmonic, sub-and superharmonic functions; polar sets, capacity and fine topology, the Dirichlet problem, Green function and green space, the general classical convergence theorem, the principle of energy and Marting boundaries

Differential equations

Differential equations as means of expressing mathematically natural laws and physical phenomena.

Ordinary differential equations, e.g. problems in mechanics, control theory,chemical kinetics, and physiology.

Partial differential equations e.g. Poisson’s equation, Laplace’s equation, the wave equation, the diffusion equation, the Fokker-Planck equation, birth and death equations, Poincare’s equation, telegraphy equation, Schr;dinger’s equation.

Special functions that arise as solutions to differential equations; e.g. the hypergeometric function, Legendre polynomials, spherical harmonics, Bessel functions.

Functional Analysis

Studies infinite-dimensional vector spaces and views functions as points in these spaces. linear functional analysis and the theory of distributions, geometric interpretation of functional analysis, Hilbert space and spectral analysis, technique of linearisation, ergodic theory, functional algebra, the Neumann-Murray theory, nonlinear functional analysis.

Calculus of variations.

Generalise functions, the theory of distribution.

Fourier Analysis

Theory of series

Fourier series

Harmonic analysis and integral transforms

Representations of groups and algebras: Fourier analysis on non-Abelian groups.

Theory of Probability

Probability of finite dimensial spaces, statistical independence.

Probability of infinite dimensional spaces; stochastic process theory, random variables. Markovian and non-Markovian processes.

Vector and tensor analysis

Scalars, vectors, tensors.

Vector algebra and analysis

Tensor algebra and analysis, including geodesics and null lines, parallel displacement of vetors, Riemann’s four-index symbols and curvature.

Topology

General Topology

Topological spaces, Euclidean n-dimensional space, Hilbert space, Cartesian-product space

Topological properties, Mappings aand their effect on topological properties: homeomorphisms.

Topological Groups and differential topology

Theorems of Tikhonov and Ascoli, continuous groups; Lie groups; Algebraic linear groups; invariants of an exgterior differential system; Poincare invariant integral and Cartan’s associate form; Poisson parentheses and contact transformations

Analysis on manifolds, Morse theory and Hodge-de Rham theory

Differential topology, the notion of a jet, embedded manifolds and intrisinc manifolds

Algebraic topology

Basic concepts of toopological spaces and maps; invariants, unchanging quantities that play a central role in the classificatoin of maps; Homotopy theory; Homology and cohomology theory: definition of a simplex; Homotopy groups; definition of fibres, fibre bundles and fibrings, sheaf cohomology; Spectral sequences, Serre, Rotheberg-Steenrod, and Eilenberg-Moore spectral sequences.

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